**1. Introduction**

274 Recent Advances in Crystallography

Publishing Company

[28] Sands D E (1982) *Vectors and tensors in crystallography*, Massachusetts: Addison-Wesley

There are two ways to the analysis of electron density distribution (EDD) measured by Xray diffraction. Multipole refinement (Stewart, 1969, 1973; Hansen & Coppens, 1978) expresses EDD as a linear combination of spherical harmonics. Using the determined functions, EDD is analyzed with Bader topological analysis (Bader, 1994). The concept of critical points of Bader theory defines the characters of chemical bonds. However, it does not give atomic orbital (AO) and its electron population, except for orbitals in high symmetry crystal fields where AO's are defined by theory. The other is based on quantummechanical orbital-models and gives the physical quantities just mentioned. The objective of this chapter is to introduce one of the second methods, the X-ray atomic orbital analysis (XAO) (Tanaka et al., 2008; hereinafter referred to as *I*), and its application to metal and rareearth complexes, where advantages of XAO are revealed quite clear.

XAO is closely related to the electron population analysis (Stewart, 1969; Coppens et al., 1971). However, it was abandoned since the orthonormal relationships between orbital functions caused severe parameter interactions in conventional least-squares refinements. Later, the method incorporating the ortho-normal relationships between AO or MO (molecular orbital) was formulated by employing Lagrange's unknown multiplier method (Tanaka, 1988; hereinafter referred to as *II).* 

The AO-based EDD refinement has been started from 3d-transition metal complexes with so high crystal symmetry that the AO's are known by the crystal field theory. Spin states of the metals in perovskites, KCoF3 and KMnF3 (Kijima et al., 1981, 1983) and KFeF3 (Miyata et al., 1983) were determined to be high spin. On the other hand, mixed orbitals of 3 2 2 *x y d* and

3 <sup>2</sup> *<sup>z</sup> d* in Jahn-Teller distorted KCuF3 were determined (Tanaka et al., 1979). The hybrid

orbital of 3 <sup>2</sup> *<sup>z</sup> d* and 4s of Cu+ which made O-Cu-O straight bond in CuAlO2 was also determined (Ishiguro et al., 1983). The investigations on KNiF3 and KCuF3 are, to our knowledge, the first ones that determined spin states and AO's by X-ray diffraction, respectively. The Cu2+ ion in [Cu(daco)2](NO3)2 (daco: diazacyclooctane) is in the crystal field *Ci*. Five d-orbitals were determined with the least-squares method stated in *II.* 

These investigations revealed that the anharmonic vibration (AHV) of atoms could not be ignored, since significant peaks still remained after removing the *3d*-peaks on difference density maps. They were attributed to the AHV of the metal atoms (Tanaka & Marumo, 1982; Ishiguro et al. 1983). For these investigations, a method proposed by Dawson et al., (1967), which had been applied only to high symmetry crystals, was made applicable to atoms in a general crystal field (Tanaka & Marumo, 1983).

Since the ratio of the number of bonding electrons to those in the unit cell becomes smaller as the atomic number increases, very accurate structure factors are necessary for X-ray EDD investigations of rare-earth compounds. Thus, the EDD analysis based on chemical-bond theories had not been done when we started the study on CeB6. Actually, the ratio in CeB6 is 1/88, which demands us to measure structure factors with the accuracy less than 1 %. Rareearth crystals are usually very hard and good resulting in enhanced extinction and multiple diffraction (MD). Therefore, MD was investigated using the method by Tanaka & Saito (1975) in which an effective way to detect MD and correct for it were proposed. It introduced the time-lag between the relevant reflections, which usually do not occur at exactly the same time, to the method by Moon and Shull (1964). The effect of MD was demonstrated through the study of PtP2 (Tanaka, et al., 1994) by measuring intensities avoiding MD and compared to those measured at the bisecting positions.

The frontier investigations aiming to measure 4f-EDD were done for CeB6 (Sato, 1985) at 100 and 298 K and for nonaaqualanthanoids (Ln:La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Yb and Lu) (Chatterjee at al. 1988). Significant peaks were found around these rare-earth atoms on the residual density maps. However, the aspherical features of 4f-EDD were not analyzed quantitatively by the X-ray scattering factors calculated with AO's in a crystal field. The first *4f*-EDD analysis of CeB6 with the crystal field model but without spin-orbit interaction exhibited *T1u* state was occupied (Tanaka et al. 1997). The ground state of the 4f-state was found to be *4f(j=5/2)* by the inelastic neutron scattering (Zirngiebl et al., 1984). When the spin-orbit interaction is taken into account, the total sum **J** of the orbital and spin angular momentum **L** and **S**, **J=L+S**, becomes a good quantum number. In the following discussion *J*  values of *p, d, f* orbitals are attached as a subscript on the right-hand side of each orbital. The experiment does not contradict to our results. It was, however, investigated again at 100, 165, 230 and 298 K (Tanaka & Ōnuki, 2002) by introducing the spin-orbit interaction. Our experiments ascertained that *4f* electron occupied *4f5/2*. orbitals. It revealed further that the electron population of them, *n(4f5/2 )*, decreased on lowering the temperature. *4f5/2* electrons are transferred to the *B6* moiety below room temperature. In addition to it just significant amount of *4f5/2*electrons were found at 298 K, which was expected to be due to the thermal excitation. Thus, the electron population above room temperature became very interesting. Since the energy gap between the two *4f5/2* orbitals is about 500 K, the X-ray intensities were measured at 430 K expecting for *4f5/2* to be more populated (Makita et al., 2007). The results were surprising. The *4f5/2* is more populated than *4f5/2*, and *5d5/2* is fully occupied at 430 K, which is expected to be at least a few ten thousands degrees higher than the *4f* orbitals. The electron configurations of CeB6 at 340 and 535 K were further investigated to confirm the results (Makita et al., 2008). The EDD of SmB6 (4f5) was studied (Funahashi, et al., 2010) below room temperature. *4f5/2*and *4f5/2* orbitals are fully or partially occupied. It is noted that *5d5/2* occupation was also found. The physical properties such as electric resistivity are correlated to the electron configuration.

### **2. Basic formalism of XAO**

#### **2.1. AO's in crystal fields**

276 Recent Advances in Crystallography

found to be *4f(j=5/2)*

amount of *4f5/2*

electron population of them, *n(4f5/2*

experiments ascertained that *4f* electron occupied *4f5/2*

orbital of 3 <sup>2</sup> *<sup>z</sup> d* and 4s of Cu+ which made O-Cu-O straight bond in CuAlO2 was also determined (Ishiguro et al., 1983). The investigations on KNiF3 and KCuF3 are, to our knowledge, the first ones that determined spin states and AO's by X-ray diffraction, respectively. The Cu2+ ion in [Cu(daco)2](NO3)2 (daco: diazacyclooctane) is in the crystal field

These investigations revealed that the anharmonic vibration (AHV) of atoms could not be ignored, since significant peaks still remained after removing the *3d*-peaks on difference density maps. They were attributed to the AHV of the metal atoms (Tanaka & Marumo, 1982; Ishiguro et al. 1983). For these investigations, a method proposed by Dawson et al., (1967), which had been applied only to high symmetry crystals, was made applicable to

Since the ratio of the number of bonding electrons to those in the unit cell becomes smaller as the atomic number increases, very accurate structure factors are necessary for X-ray EDD investigations of rare-earth compounds. Thus, the EDD analysis based on chemical-bond theories had not been done when we started the study on CeB6. Actually, the ratio in CeB6 is 1/88, which demands us to measure structure factors with the accuracy less than 1 %. Rareearth crystals are usually very hard and good resulting in enhanced extinction and multiple diffraction (MD). Therefore, MD was investigated using the method by Tanaka & Saito (1975) in which an effective way to detect MD and correct for it were proposed. It introduced the time-lag between the relevant reflections, which usually do not occur at exactly the same time, to the method by Moon and Shull (1964). The effect of MD was demonstrated through the study of PtP2 (Tanaka, et al., 1994) by measuring intensities

The frontier investigations aiming to measure 4f-EDD were done for CeB6 (Sato, 1985) at 100 and 298 K and for nonaaqualanthanoids (Ln:La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Yb and Lu) (Chatterjee at al. 1988). Significant peaks were found around these rare-earth atoms on the residual density maps. However, the aspherical features of 4f-EDD were not analyzed quantitatively by the X-ray scattering factors calculated with AO's in a crystal field. The first *4f*-EDD analysis of CeB6 with the crystal field model but without spin-orbit interaction exhibited *T1u* state was occupied (Tanaka et al. 1997). The ground state of the 4f-state was

spin-orbit interaction is taken into account, the total sum **J** of the orbital and spin angular momentum **L** and **S**, **J=L+S**, becomes a good quantum number. In the following discussion *J*  values of *p, d, f* orbitals are attached as a subscript on the right-hand side of each orbital. The experiment does not contradict to our results. It was, however, investigated again at 100, 165, 230 and 298 K (Tanaka & Ōnuki, 2002) by introducing the spin-orbit interaction. Our

are transferred to the *B6* moiety below room temperature. In addition to it just significant

by the inelastic neutron scattering (Zirngiebl et al., 1984). When the

electrons were found at 298 K, which was expected to be due to the thermal

 *)*, decreased on lowering the temperature. *4f5/2*

. orbitals. It revealed further that the

electrons

*Ci*. Five d-orbitals were determined with the least-squares method stated in *II.* 

avoiding MD and compared to those measured at the bisecting positions.

atoms in a general crystal field (Tanaka & Marumo, 1983).

The i-th AO, , r *<sup>i</sup>* (i=1,2,...,M) of the -th atom are expressed with N basisfunctions, ( ) *<sup>k</sup>* **r** ,

$$\mathcal{W}\_{\alpha,i}(r) := \sum\_{k}^{N} c\_{ik} \mathcal{W}\_{k}(\mathbf{r}), \ i = 1, 2, \dots, M,\tag{1}$$

where *cik* is a constant to be determined in the XAO analysis with the least-squares method incorporating orthonormal relationship between AO's (Tanaka, 1988). Matrices and vectors are written in upper-case letters and bold lower-case letters, and superscript ' means a transposed matrix or a row vector, respectively. The basis functions, ( ) *<sup>k</sup>* **r** are listed in Table 1 of *I*. They are expressed using polar coordinate *(r, ,*  as a linear combination of a product of a radial function *Rnl(r)* and a spherical harmonic (,) *l Ylm* as,

$$\Psi\nu\_k(\mathbf{r}) = \sum\_{m\_l=-l}^{l} d\_{k,m\_l} R\_{nl}(r) Y\_{lm\_l}(\theta\_\prime \phi)\_\prime \tag{2}$$

where *n*, *l* and *ml* are principal, azimuthal and magnetic quantum numbers. The nonrelativistic radial functions, *Rnl(r)* (Mann, 1968) and relativistic functions calculated by *HEX* (Liberman et al., 1971) were mainly used for the XAO analysis. , *<sup>l</sup> k m <sup>d</sup>* 's are listed in Table 1 of I. Since approximate *cik'*s were necessary at the start of the non-linear least-squares calculation, they were calculated by taking the crystal field, which was calculated by placing a proper point charge on each atom, as a perturbation to the system (Kamimura, et al., 1969). For details, see eqs. (52)-(56) of *I*. The adjustable variables *cik*'s in the least-squares calculation are listed for all point group symmetries in Tables 4 and 5 in *I.* 

#### **2.2. Least-squares method incorporating ortho-normal condition**

In a non-linear least-squares refinement, mathematical and physical relationships between parameters should be taken into account to avoid parameter interaction. The conventional one used in X-ray crystallography was improved to obtain AO/MO by taking into account the orthonormal relationship between wave functions (Tanaka, 1988),

$$\sum\_{m}^{N} \sum\_{n}^{N} c\_{im}^{\*} c\_{jn} s\_{mn} = \mathcal{S}\_{ij} \tag{3}$$

where *mn m n s d* **<sup>r</sup>** and *N* is the number of basis functions. *δij* is Kronecker's *δ*. Assuming cim's are real, it is rewritten with *(M,N)* matrix *C={cim}* and *(N,N)* matrix *S={smn}*,

$$\text{CSC}^{\prime} = I\_{\prime} \tag{4}$$

where *I* is the unit matrix of order *M.* Using Lagrange's unknown multiplier *λij*,, the value *Q* to be minimized in the least-squares method becomes,

$$\mathbf{Q} = \mathbf{v}' \mathbf{M}\_f^{-1} \mathbf{v} - \sum\_{i \le \quad j}^{M} \sum\_{j}^{M} \lambda\_{ij} (\sum\_{m}^{N} \sum\_{n}^{N} c\_{im} c\_{jn} \mathbf{s}\_{mn} - \delta\_{ij}) \tag{5}$$

For definition of **v** see eqs. (10a) to (10d) of II. Putting *cim* as the sum of the true value <sup>0</sup> *imc* and its small shift *cim*, *Q* is expressed by ignoring the terms higher than second order of *cim*,

$$Q = \mathbf{v}' M\_f^{-1} \mathbf{v} - \sum\_{i \le \quad j}^{M} \sum\_{j}^{M} \mathcal{A}\_{ij} (\sum\_{m}^{N} \sum\_{n}^{N} \{\Delta c\_{im} c\_{jn}^{0} + c\_{im}^{0} c\_{jn}\} s\_{mn} - \mathcal{S}\_{ij}),\tag{6}$$

Putting ij'ijij Q is differentiated to give,

$$\delta \mathcal{Q} = 2 \mathfrak{ds}' \langle A'M\_f^{-1} A \mathbf{x} - A'M\_f^{-1} \mathbf{f} \rangle - 2 \sum\_{i}^{M} \sum\_{j}^{M} \lambda\_{ij}^{\*} \sum\_{m}^{N} \sum\_{n}^{N} \delta \{\Delta c\_{im} c\_{jn}^{0} s\_{mn} \} = \mathbf{0},\tag{7}$$

where **x** represents least-squares estimates of parameters. The first term is obtained following the usual procedure. For **x**, *A* and *Mf*, see Hamilton (1964). After alligning *cim*'s linearly and adding at the end of **x** for simplicity, *ij* is expressed in terms of *ci*m by using the orthonormal relationship in eq. (3). Then the final secular equation is obtained,

$$(I - R)A'M\_f^{-1}A\mathbf{x} = (I - R)A'M\_f^{-1}\mathbf{f},\tag{8}$$

where *I* is unit matrix of order *P*, total number of variables. The (P,P) matrix *R* is expressed in terms of *cik*'s. It makes the inverse matrix of <sup>1</sup> ( )' *<sup>f</sup> I RAM A* in (8) calculable. The explicit form of *R={rij}* under the limiting conditions (10) for *i, j, k* and *l* is,

$$\mathbf{r}\_{\mathrm{ij}} = \left< \mathbf{S} \mathbf{C}' \mathbf{C} \right>\_{\mathrm{kl}},\tag{9}$$

$$\begin{aligned} \text{P}-\text{MN} + \left(\text{m}-1\right)\text{N} + 1 &\stackrel{<}{\text{i}} \text{j} \stackrel{<}{\le} \text{P}-\text{MN} + \text{mN},\\ \text{k} = \text{i} - \left\{\text{P}-\text{MN} + \left(\text{m}-1\right)\text{N}\right\}, &\text{l} = \text{j} - \left\{\text{P}-\text{MN} + \left(\text{m}-1\right)\text{N}\right\}. \end{aligned} \tag{10}$$

where m runs from 1 to M. The other elements *rij* not defined here are zero.

#### **2.3. Electron density and structure factors**

278 Recent Advances in Crystallography

where *mn m n s d* 

its small shift

one used in X-ray crystallography was improved to obtain AO/MO by taking into account

*ccs*

*im jn mn ij*

' *CSC I* , (4)

<sup>1</sup> '( ) *MM N N*

*f ij im jn mn ij i j mn*

*cim*, *Q* is expressed by ignoring the terms higher than second order of

*f ij im jn im jn mn ij*

*cc cc s*

11 ' 0 2 '( ' ' ) 2 ( ) 0,

**v v** (6)

*MM N N f f ij im jn mn i j mn*

*ccs* **<sup>δ</sup>x xf** (7)

 

ij kl r SC'C , (9)

*ccs*

*ij* is expressed in terms of *ci*m by using the

**v v** (5)

where *I* is the unit matrix of order *M.* Using Lagrange's unknown multiplier *λij*,, the value *Q*

 <sup>1</sup> 0 0 ' ( ( ) ), *MM N N*

where **x** represents least-squares estimates of parameters. The first term is obtained following the usual procedure. For **x**, *A* and *Mf*, see Hamilton (1964). After alligning

1 1 ( )' ( )' , *f f I RAM A I RAM* **x f** (8)

where *I* is unit matrix of order *P*, total number of variables. The (P,P) matrix *R* is expressed in terms of *cik*'s. It makes the inverse matrix of <sup>1</sup> ( )' *<sup>f</sup> I RAM A* in (8) calculable. The explicit

k i P MN m 1 N , l j P MN m 1 N ,

 

For definition of **v** see eqs. (10a) to (10d) of II. Putting *cim* as the sum of the true value <sup>0</sup>

*i j mn*

orthonormal relationship in eq. (3). Then the final secular equation is obtained,

P MN m 1 N 1 i,j P MN mN,

*Q AM A AM*

form of *R={rij}* under the limiting conditions (10) for *i, j, k* and *l* is,

**<sup>r</sup>** and *N* is the number of basis functions. *δij* is Kronecker's *δ*. Assuming

(3)

*imc* and

*cim*'s

(10)

*cim*,

the orthonormal relationship between wave functions (Tanaka, 1988),

to be minimized in the least-squares method becomes,

*Q M*

Putting ij'ijij Q is differentiated to give,

linearly and adding at the end of **x** for simplicity,

*Q M*

\* *N N*

*m n*

cim's are real, it is rewritten with *(M,N)* matrix *C={cim}* and *(N,N)* matrix *S={smn}*,

The electron density *(***r***)* of the αth atom centered at *atom* **r** is divided into that of the core and valence orbitals, , ( ) *a valence* **r** . It is further expressed from eq. (1),

$$\boldsymbol{\rho}\_{\boldsymbol{\alpha},\boldsymbol{\text{valence}}}(\mathbf{r}) = \sum\_{i} \boldsymbol{n}\_{\boldsymbol{\alpha},i} \boldsymbol{\rho}\_{\boldsymbol{\alpha},i} (\boldsymbol{\kappa}\_{i} (\mathbf{r} - \mathbf{r}\_{\boldsymbol{\alpha}}^{\rm atom})) = \sum\_{i} \boldsymbol{n}\_{\boldsymbol{\alpha},i} \boldsymbol{\Psi}\_{\boldsymbol{\alpha},i}^{\*} (\boldsymbol{\kappa}\_{i} (\mathbf{r} - \mathbf{r}\_{\boldsymbol{\alpha}}^{\rm atom})) \boldsymbol{\Psi}\_{\boldsymbol{\alpha},i} (\boldsymbol{\kappa}\_{i} (\mathbf{r} - \mathbf{r}\_{\boldsymbol{\alpha}}^{\rm atom})), \tag{11}$$

where ,*<sup>i</sup> n* is the number of electrons of ,*<sup>i</sup>* . *<sup>i</sup>* expresses the expansion 1 *<sup>i</sup>* or contraction 1 *<sup>i</sup>* of i-th orbital (Coppens et al., 1979). Atom α in the asymmetric unit is translated to *atom* **r** by the σ-th crystal symmetry operation. Then the structure factor is,

$$F(\mathbf{k}) = \sum\_{a} \alpha\_{a} \sum\_{\sigma} f\_{a\sigma}(\mathbf{k}) \exp[i\mathbf{k} \cdot \mathbf{r}\_{a\sigma}^{atom}] T\_{a\sigma}(\mathbf{k}) \,\tag{12}$$

where ( ) *f* **k** is an atomic scattering factor and **k** is a scattering vector. The *α*th atom at *atom* **r** is displaced by atomic vibration to *atom* **r u** . It perturbs the periodicity of the crystal and reduces diffracted intensity. Then the atomic displacement parameter (ADP),

$$T\_{a\sigma}^{atom}(k) = \left\langle \exp(i\mathbf{k}\cdot\mathbf{u}) \right\rangle,\tag{13}$$

is calculated as an average in space and time of exp(i**k**·**u**) by assuming each atom vibrates independently from each other. *f* is the sum of the scattering factors of core and valence electrons, ( ) *core f* **k** and ( ) *valence f* **k** . ( ) *valence f* **k** is expressed as,

$$f\_{\alpha\sigma}^{valence}(\mathbf{k}) = \sum\_{i} n\_{i} f\_{\alpha,\sigma,i}^{valence}(\mathbf{k})\_{\prime} \tag{14}$$

where *ni* is the number of electrons of the *i*th valence orbitals. , , ( ) *valence <sup>i</sup> f* **k** is a Fourier transform of , , ( ) *valence i* **r** . Final expression of , , *valence <sup>i</sup> f* becomes,

$$\begin{split} f\_{\alpha,\sigma,i}^{\text{volume}} &= \sum\_{k} \sum\_{k'} c\_{ik}^{\star} c\_{ik} \sum\_{K=ll-l'\uparrow}^{l+l'} < j\_K > \sum\_{M=-K}^{K} i^K \sqrt{2(2K+1)} \Theta\_K^M(\beta\_\sigma) \exp(iM\gamma\_\sigma) \\ &\times \sum\_{m\_l=-1}^{l} \sum\_{m\_l=-l'}^{l'} d\_{k,m\_l}^{\star} d\_{k',m\_l} c^K \langle lm\_l, l'm\_{l'}\rangle, \end{split} \tag{15}$$

where *K+l+l'* is even. '' '. *M m m and l l K l l j l l <sup>K</sup>* is expressed as,

$$
\left\langle j\_K \right\rangle = \int R\_{nl}(r) R\_{n'l'}(r) j\_K(kr) r^2 dr \tag{16}
$$

*jK* is Bessel function of order *K*. For *(*, see Appendix B of *I*.

#### **2.4. Anharmonic vibration (AHV)**

Gaussian probability density functions (p.d.f.) of atoms are Fourier transformed to give harmonic ADP. Gram-Charier (G-C) formalism is widely used to introduce AHV. However, it expresses aspherical EDDs both from orbital functions and AHV, as shown clearly by Mallinson et al. (1988). Thus, it may not be adequate for accurate EDD researches. The method based on Boltzmann statistics (Dawson et al. 1967, Willis, 1969) was employed,

$$T\_{\alpha\sigma}^{\text{atom}}(\mathbf{k}) = \left[ \exp(-V(\mathbf{u}) / k\_B T) \exp(i \mathbf{k} \cdot \mathbf{u}) d\mathbf{u} \right] \left[ \exp(-V(\mathbf{u}) / k\_B T) d\mathbf{u},\tag{17}$$

where *kB* is Boltzmann constant, and *T* is absolute temperature. Assuming each atom vibrates independently and defining *u(u1,u2,u3)* on the Cartesian coordinates parallel to the principal axes of the thermal ellipsoid of harmonic ADP, potential V(**u**) is expressed as,

$$V(\mathbf{u}) = V\_0 + \frac{1}{2} \sum\_{i=1}^3 b\_i u\_i^2 + \sum\_{i,j,k}^\cdot c\_{ijk} u\_i u\_j u\_k + \sum\_{i,j,k,l}^\cdot q\_{ijkl} u\_i u\_j u\_k u\_l + \dots \tag{18}$$

where i, j, k, l run from 1 to 3 and ' means they are permutable. There are 10 cubic and 15 quartic terms. Potential expansion terminates at the quadratic terms in the harmonic ADP.

By assuming the sum of the third and the fourth terms in (18) is much smaller than *kBT*, *bi, cijk* and *qijkl* are introduced into structure factor formalism using (12), (17) and (18).

#### **3. Experimental**

#### **3.1. Multiple diffraction (MD)**

MD occurs when two or more reflections are on the Ewald sphere. When the incident beam with wave length enters parallel to the unit vector **j,** and a primary reflection, the intensity of which is measured, is at the reflecting position, the conditions for a secondary reflection, *h1h2h3*, to occur become as follows (Tanaka & Saito, 1975),

$$\left| \mathbf{e}\_3 \cdot (\mathbf{j} / \mathcal{X} + h\_3 \mathbf{b}\_3) \right| \le 1 / \mathcal{X} \tag{19}$$

$$\left| \mathbf{e}\_2 \cdot \left( \mathbf{j} / \,\mathcal{A} + h\_2 \mathbf{b}\_2 + h\_3 \mathbf{b}\_3 \right) \right| \le \left[ 1 / \,\mathcal{A}^2 - \left\{ \mathbf{e}\_3 \cdot \left( \mathbf{j} / \,\mathcal{A} + h\_3 \mathbf{b}\_3 \right)^2 \right\}^{1/2} \right]^{1/2} = r\_3 \tag{20}$$

$$\mathbf{e}\_1 \cdot \left( \mathbf{j} / \,\% + h\_1 \mathbf{b}\_1 + h\_2 \mathbf{b}\_2 + h\_3 \mathbf{b}\_3 \right) = \pm \left[ r\_3^2 - \left\{ e\_2 \bullet \left( \mathbf{j} / \,\% + h\_2 \mathbf{b}\_2 + h\_3 \mathbf{b}\_3 \right) \right\}^2 \right] \tag{21}$$

**b***1,* **b***2,* **b***3* are reciprocal-lattice vectors when the primary reflection exactly fulfills the reflecting condition and **e***1,* **e***2,* **e***3* are the unit vectors defined as,

XAO Analysis – AO's and Their Populations in Crystal Fields 281

$$\mathbf{e}\_3 = \frac{\mathbf{b}\_1 \times \mathbf{b}\_2}{\left| \mathbf{b}\_1 \times \mathbf{b}\_2 \right|}, \mathbf{e}\_2 = \frac{\mathbf{e}\_3 \times \mathbf{b}\_1}{\left| \mathbf{e}\_3 \times \mathbf{b}\_1 \right|}, \mathbf{e}\_1 = \frac{\mathbf{b}\_1}{\left| \mathbf{b}\_1 \right|}. \tag{22}$$

Secondary reflections only around the surface of the Ewald sphere are searched using eqs. (19) to (21). Integers *h3* are selected from (19) and integers *h2* are calculated from (20) for each *h3*. Then the value of *h1* is evaluated from eq. (21) for each (*h2, h3*). The point *(h1 ,h2, h3)* is judged to be a reciprocal lattice point if *h1* is an integer. The Ewald sphere has width due to the divergence of the incident beam, finite size and mosaicity of the crystal, and wavelength spread. It permits some margin from an integer to *h1*.

280 Recent Advances in Crystallography

was employed,

**3. Experimental** 

with wave length

**3.1. Multiple diffraction (MD)** 

*jK* is Bessel function of order *K*. For *(*

**2.4. Anharmonic vibration (AHV)** 

( ) exp( ( ) / )exp( ) / exp( ( ) / ) , *atom*

2

and *qijkl* are introduced into structure factor formalism using (12), (17) and (18).

reflection, *h1h2h3*, to occur become as follows (Tanaka & Saito, 1975),

reflecting condition and **e***1,* **e***2,* **e***3* are the unit vectors defined as,

3 33 **ej b** ( / 1/

0

*B B T V kT i d V kTd*

3' '

1 ,, ,,, <sup>1</sup> ( ) ..... <sup>2</sup> *i i ijk i j k ijkl i j k l i ijk ijkl V V bu c uuu q uuu u*

where i, j, k, l run from 1 to 3 and ' means they are permutable. There are 10 cubic and 15 quartic terms. Potential expansion terminates at the quadratic terms in the harmonic ADP. By assuming the sum of the third and the fourth terms in (18) is much smaller than *kBT*, *bi, cijk*

MD occurs when two or more reflections are on the Ewald sphere. When the incident beam

intensity of which is measured, is at the reflecting position, the conditions for a secondary

2 22 33 3 33 3 ( /

*h h* 1/ ( / *h r* **ej b b ej b** (20)

*h h h re h h* ) ( / ) **ej b b b j b b** (21)

<sup>1</sup> 11 22 33 3 2 22 33 ( /

**b***1,* **b***2,* **b***3* are reciprocal-lattice vectors when the primary reflection exactly fulfills the

*h*

<sup>2</sup> <sup>2</sup>

where *kB* is Boltzmann constant, and *T* is absolute temperature. Assuming each atom vibrates independently and defining *u(u1,u2,u3)* on the Cartesian coordinates parallel to the principal axes of the thermal ellipsoid of harmonic ADP, potential V(**u**) is expressed as,

, see Appendix B of *I*.

**<sup>k</sup> u ku u u u** (17)

**<sup>u</sup>** (18)

enters parallel to the unit vector **j,** and a primary reflection, the

 1/2 <sup>2</sup> <sup>2</sup>

> 

(19)

Gaussian probability density functions (p.d.f.) of atoms are Fourier transformed to give harmonic ADP. Gram-Charier (G-C) formalism is widely used to introduce AHV. However, it expresses aspherical EDDs both from orbital functions and AHV, as shown clearly by Mallinson et al. (1988). Thus, it may not be adequate for accurate EDD researches. The method based on Boltzmann statistics (Dawson et al. 1967, Willis, 1969)

> The change of intensity by MD was first formulated by Moon & Shull (1964). Primary and secondary beams usually come on the Ewald sphere with an angle-lag . Using The formalism was modified for a spherical crystal with a radius *r* (Tanaka & Saito, 1975),

$$\frac{\Delta I\_1}{I\_1} = \frac{\sqrt{\pi r^2}}{8} Q\_{01} \sum\_i \left[ -g\_{01;0i} \left( \frac{Q\_{0i}}{Q\_{01}} \right) - g\_{01;1i} \left( \frac{Q\_{1i}}{Q\_{01}} \right) + g\_{01;i1} \left( \frac{Q\_{0i}}{Q\_{01}} \right) \left( \frac{Q\_{i1}}{Q\_{01}} \right) \right]. \tag{23}$$

Subscripts *'0', '1'* and *'i'* stand for the incident, primary and secondary beams. *Qmn* is the integrated reflectivity per unit volume of a crystal for the *m*-th incident beam and *n*-th diffracted beam (*m-n* process). It is proportional to the squares of the structure factor. In *i-1* process, for example, a secondary beam *i* acts as an incident beam and is diffracted along the direction of the primary beam *1*, which enhances the intensity of the primary beam. *gij;mn* is a function of Lorentz and polarization factors of the simultaneous *i-j* and *m-n* processes. The polarization factor for a general multi-diffracted process was recently formulated (Tanaka et al., 2010). For *gij;mn* see Tanaka & Saito (1975). Since *I*1*/I*1 depends on the ratio of *Qmn*'s and of sharp primary and secondary reflections,  *I*1 fluctuates sharply. Eq. (23) was applied to diformohydrazide (Tanaka, 1978) and two peaks on the N-N bond in the difference density map became one peak at the middle of the bond.

The EDD analyses of compounds with metals and rare-earth elements need accurate measurements of structure factors, however MD affects the measured intensity seriously since *Qmn* is large. However correction for it does not seem fruitful, since each primary reflection has many secondary reflections and MD depends sharply on crystal orientation and half-width of each Bragg peak. Thus it is better to avoid MD by rotating a crystal around the reciprocal vector of the primary reflection ( rotation) so that secondary reflections move apart from the Ewald sphere and measure at where *I1/I1* is minimum. For actinoid compounds, MD cannot be avoided and the correction for it becomes necessary.

#### **4. XAO analysis for crystals in the** *Oh* **crystal field**

#### **4.1. 3d, 4f and 5d orbitals in the** *Oh* **crystal field**

In the present chapter EDD investigations on the first-transition metal complexes and rare earth compounds in *Oh* crystal field are mainly stated. Thus it is worth while to represent the energy level splitting of d- and f-states in Figs.1(a) and (b) (Funahashi, 2010). The spin-orbit splitting is neglected in a strong field model. For *(3d)n* systems strong field model is often employed. In *(4f)n* and *(5d)n* systems the two splittings are of the same order of magnitude. Weak field models are employed for CeB6 and SmB6. Their quantization axes point to the face-centre of the cubic unit cell, where no B atom exists. Note that the order of the levels in Fig. 1(a) is inverted, that is, t2g orbitals lies lower than eg, when ligands are on the quantization axes like KCoF3 (section 4). In weak field *5d*- and *4f*-states split into *j*=3/2 and *j*=5/2 states, and into *j=5/2* and *j=7/2* states by the spin-orbit interaction. The *Oh* crystal field splits them further. Lobes of *4f5/28* extend along <100> directions while those of *4f5/27* orbitals along <110> and <111> as illustrated in Fig. 7 of Makita et al. (2007).

**Figure 1.** Energy-level splitting of (a) d- and (b) f-states in the *Oh* crystal field with the degeneracy of each state in parentheses. Each orbital of a strong field can have two electrons.

#### **4.2.** *3d***-EDDand spin states in KCoF3, KMnF3 and KFeF3**

The peak due to 3d electrons was first reported for [Co(NH3)6][Co(CN)6] (Iwata & Saito, 1975) around the Co atom in 3 crystal field. Later electron distribution at 80 K was reported (Iwata, 1977). *3d*-EDD in Co2SiO4 (Marumo et al., 1977) and CoAl2O4 (Toriumi, et al., 1978) were also observed. The electron density around Co in KCoF3 was first analysed quantitatively based on the *3d*-orbitals in *Oh* crystal field (Kijima et al., 1981). A crystal was shaped into a sphere of diameter 0.13 mm, and intensities were measured up to 2°with a four-circle diffractometer using MoK radiation without avoiding MD. *cik'*s in (1) are fixed constants because of the high symmetry as listed in Tables 1(b) and 5(b) of *I*. *<sup>i</sup>* in eq. (11) was not introduced to the investigations in sections 4 5.1 and 5.2. Scattering factors of 22 2 ,,, *xy z yz zx xy d d d d and d* orbitals are calculated with eq. (15). When electrons occupy the five *d-*orbitals equally, that is *(t2g)4.2(eg)2.8* for Co2+, atoms have spherical EDD. Hereinafter the refinement with the spherical scattering factors and without assuming AHV is called as spherical-atom refinement, in which type *I* or *II* anisotropic extinction (Becker & Coppens, 1974a, b, 1975) was assumed. The difference density map assuming the sphericalatom, high and low spins, *(t2g)5(eg)2* and *(t2g)6(eg)1,* are shown in Figs. 2(a) to (c). Peaks remained after spherical-atom refinement were reduced and enhanced after high- and lowspin refinements, respectively. Large and almost no peaks in Fig. 2(c) and Fig. 2(b) demonstrate that Co2+ is in the high-spin state. Spin-state was first determined by X-ray diffraction in this study. In the similar way, the spin-states of 3d-transition metals in KMnF3 (Kijima et al., 1983) and KFeF3 (Miyata et al., 1983) were determined to be high-spin.

**Figure 2.** Difference density after refinements of (a) spherical-atom, (b) high-spin, (c) low- spin models. Contours at 0.2 eÅ-3. Negative and zero in broken and dashed-dotted lines.

#### **5. Experimental AO determination**

282 Recent Advances in Crystallography

splits them further. Lobes of *4f5/2*

energy level splitting of d- and f-states in Figs.1(a) and (b) (Funahashi, 2010). The spin-orbit splitting is neglected in a strong field model. For *(3d)n* systems strong field model is often employed. In *(4f)n* and *(5d)n* systems the two splittings are of the same order of magnitude. Weak field models are employed for CeB6 and SmB6. Their quantization axes point to the face-centre of the cubic unit cell, where no B atom exists. Note that the order of the levels in Fig. 1(a) is inverted, that is, t2g orbitals lies lower than eg, when ligands are on the quantization axes like KCoF3 (section 4). In weak field *5d*- and *4f*-states split into *j*=3/2 and *j*=5/2 states, and into *j=5/2* and *j=7/2* states by the spin-orbit interaction. The *Oh* crystal field

*8* extend along <100> directions while those of *4f5/2*

*7*

*<sup>i</sup>* in eq. (11)

orbitals along <110> and <111> as illustrated in Fig. 7 of Makita et al. (2007).

**Figure 1.** Energy-level splitting of (a) d- and (b) f-states in the *Oh* crystal field with the degeneracy of

The peak due to 3d electrons was first reported for [Co(NH3)6][Co(CN)6] (Iwata & Saito, 1975) around the Co atom in 3 crystal field. Later electron distribution at 80 K was reported (Iwata, 1977). *3d*-EDD in Co2SiO4 (Marumo et al., 1977) and CoAl2O4 (Toriumi, et al., 1978) were also observed. The electron density around Co in KCoF3 was first analysed quantitatively based on the *3d*-orbitals in *Oh* crystal field (Kijima et al., 1981). A crystal was shaped into a sphere of diameter 0.13 mm, and intensities were measured up to 2°with a four-circle diffractometer using MoK radiation without avoiding MD. *cik'*s in (1) are fixed

was not introduced to the investigations in sections 4 5.1 and 5.2. Scattering factors of 22 2 ,,, *xy z yz zx xy d d d d and d* orbitals are calculated with eq. (15). When electrons occupy the five *d-*orbitals equally, that is *(t2g)4.2(eg)2.8* for Co2+, atoms have spherical EDD. Hereinafter the refinement with the spherical scattering factors and without assuming AHV is called as spherical-atom refinement, in which type *I* or *II* anisotropic extinction (Becker &

constants because of the high symmetry as listed in Tables 1(b) and 5(b) of *I*.

each state in parentheses. Each orbital of a strong field can have two electrons.

**4.2.** *3d***-EDDand spin states in KCoF3, KMnF3 and KFeF3** 

When symmetries of crystal fields are low, *cik'*s in (1) become unknown variables, and it is necessary to determine them by the least-squares method keeping the orthonormal relationship between AO's. Independent variables, cik's, and the relation between them are listed for 32 point group symmetries in Tables 4 and 5 of *I*.

#### **5.1. Jahn-Teller distortion in KCuF3 and mixed** 2 2 *x y <sup>d</sup>*  **and** <sup>2</sup> *<sup>z</sup> <sup>d</sup>* **orbital**

In KCuF3 each F ion between Cu2+ ions (3d9) shifts from the centre by Jahn-Teller effect. It makes short Cu-Fs, medium Cu-Fm, parallel to *c*-axis, and long Cu-Fl bonds, resulting in *mmm* point group symmetry for Cu2+. Difference density after spherical-atom refinement exhibits non-equivalent four holes in Fig. 3 (b) (Tanaka et al. 1979). Table 5(b) of *I* indicates a mixing of 3 3 <sup>2</sup> 2 2 *<sup>z</sup> x y d and d* in the crystal field *mmm*,

$$\begin{aligned} \left. \psi \right|\_{G} &= \cos(\wp / 2) d\_{\underline{z}^2} + \sin(\wp / 2) d\_{\underline{x}^2 - \underline{y}^2}, \\ \left. \psi \right|\_{\underline{x}} &= \sin(\wp / 2) d\_{\underline{z}^2} - \cos(\wp / 2) d\_{\underline{x}^2 - \underline{y}^2}. \end{aligned} \tag{24}$$

The peaks on *Cu-Fl* in Fig. 3(a) correspond to the lone pair of *<sup>G</sup>* . Putting two and one electrons to *<sup>G</sup>* and *<sup>E</sup>* , *cos(/2)* became 0.964(18). 3d-peaks in Fig. 3 (a) and (b) were deleted in (c) and (d). It confirms orbitals in eq. (24) are quite reasonable. The orbital functions except the phase factor were determined for the first time from X-ray diffraction experiment. However, significant peaks still remained. The positive and negative peaks along <100> and <110> in Fig.3(c) indicate preferential and inhibitive vibration of Cu2+, respectively. Therefore, AHV of Cu2+ was analysed (Tanaka & Marumo, 1982). The obtained potential in (18) corresponds to the peaks and removes them as seen in the residual density in Fig. 4. Our study on AHV started from this investigation. Since the AHV peaks appear in Fig. 3(c) and (d) after the *d-*electron peaks were removed, the aspherical peaks due to electron configuration and vibration of Cu2+ seem to be separated well.

**Figure 3.** Difference density after spherical-atom refinement on (a) (001) and (b) (220) planes passing Cu2+ at (1/2,0,0). That after d-orbital analysis on (c) (001) and (d) (220). Contours are the same as those in Fig. 2.

**Figure 4.** Difference density after AHV analysis. Contours are the same as those in Fig.2.

#### **5.2. 3d orbitals in** 1 **crystal field of Cu(daco)2(NO3)2**

The Cu2+ ion (3d9) of Cu(daco)2(NO3)2(daco: 1,5-diazacyclooctane) is on the centre of symmetry and forms a coordination plane with the four N atoms in daco (Hoshino, et al., 1989). Difference densities on and perpendicular to the Cu-N4 plane after the spherical-atom refinement are shown in Fig. 5(a) and (b). Negative four peaks in Fig. 5 (a) near Cu2+ correspond to a 3d-hole. However, the peaks do not locate on the Cu-N bonds. Since the crystal field of the Cu atom is Ci, d-orbitals of Cu2+ are expressed as a linear combination of 22 2 , , , *xy z yz zx xy d d d d and d* orbitals and all the 25 coefficients cik were refined with the method in *II*. Since approximate cik's are necessary for the non-linear least-squares refinement, they are calculated putting a point charge on each atom (Kamimura et al., 1969). *<sup>K</sup>* <sup>3</sup>*<sup>d</sup> <sup>j</sup>* of Cu2+ were taken from International Tables for X-ray crystallography (1974, vol. IV). After each cycle of refinement, new set of coefficients were orthonormalized by the Löwdin's method (Löwdin, 1950). Refinement was converged and the coefficients cik's, population *ni* and κi are listed in Table 1. The number of significant cik's is seven among 25 coefficients. Orbital 1 in Table 1 is the hole orbital. It is not a pure 2 2 *x y d* but a mixed orbital of 2 2 *x y d* and *zx d* orbitals, *c14*=0.57(59) is not significant. The hole and large negative and positive peaks in Fig. 5 were removed in Fig. 6. Further AHV refinement removed the peaks in Fig.6, but that prior to the d-orbital analysis did not improve Fig. 5.

284 Recent Advances in Crystallography

Fig. 2.

deleted in (c) and (d). It confirms orbitals in eq. (24) are quite reasonable. The orbital functions except the phase factor were determined for the first time from X-ray diffraction experiment. However, significant peaks still remained. The positive and negative peaks along <100> and <110> in Fig.3(c) indicate preferential and inhibitive vibration of Cu2+, respectively. Therefore, AHV of Cu2+ was analysed (Tanaka & Marumo, 1982). The obtained potential in (18) corresponds to the peaks and removes them as seen in the residual density in Fig. 4. Our study on AHV started from this investigation. Since the AHV peaks appear in Fig. 3(c) and (d) after the *d-*electron peaks were removed, the aspherical peaks due to

**Figure 3.** Difference density after spherical-atom refinement on (a) (001) and (b) (220) planes passing Cu2+ at (1/2,0,0). That after d-orbital analysis on (c) (001) and (d) (220). Contours are the same as those in

**Figure 4.** Difference density after AHV analysis. Contours are the same as those in Fig.2.

The Cu2+ ion (3d9) of Cu(daco)2(NO3)2(daco: 1,5-diazacyclooctane) is on the centre of symmetry and forms a coordination plane with the four N atoms in daco (Hoshino, et al., 1989). Difference densities on and perpendicular to the Cu-N4 plane after the spherical-atom refinement are shown in Fig. 5(a) and (b). Negative four peaks in Fig. 5 (a) near Cu2+ correspond to a 3d-hole. However, the peaks do not locate on the Cu-N bonds. Since the crystal field of the Cu atom is Ci, d-orbitals of Cu2+ are expressed as a linear combination of 22 2 , , , *xy z yz zx xy d d d d and d* orbitals and all the 25 coefficients cik were refined with the

**5.2. 3d orbitals in** 1 **crystal field of Cu(daco)2(NO3)2** 

electron configuration and vibration of Cu2+ seem to be separated well.

**Figure 5.** Difference density after spherical-atom refinement on (a) Cu-N4 plane and (b) the plane perpendicular to it. Contours are the same as those in Fig. 2.

**Figure 6.** Difference density after 3d-orbital analysis. Contours are the same as in Fig. 2

When G-C formalism was applied to an iron complex, it removed the peaks equally well as the multipole refinement did (Mallinson, et al., 1988). Our AHV analysis is based on the classical thermodynamics and does not have such a problem, though the p.d.f. function *exp[-V(***u***)/kBT]* is not integrable (Scheringer, 1977) at a place far from the nucleus. However, we can apply it safely as long as it is applied within the region where the sum of the third and fourth terms in (18) is much smaller than *kBT*. The condition is usually fulfilled in the investigations of EDD.


**Table 1.** 3d-orbital parameters cik's of Cu2+. Significant ones in thick lines
